Basic Statistics

OTHER TOPICS IN LINEAR REGRESSION 183

I

X

Figure 12.7 Shape of a nonlinear regression curve.

12.5 OTHER TOPICS IN LINEAR REGRESSION

In this section, two methods of correcting potential problems that might occur in
fitting linear regression lines are discussed briefly and references are given to more
extensive information. Also, a very brief mention of multiple regression analysis is
given.

12.5.1

Examination of the scatter diagram sometimes reveals that the points do not lie on a
straight line. In Section 6.5 we mentioned taking transformations of the data in order
to achieve a distribution that was approximately normally distributed. We discussed
the use of the logarithmic and square-root transformations if the observations were
skewed to the right. These two transformations are often used to transform a curve
into a straight line.
Figure 12.7 shows a hypothetical regression curve that is commonly found in
biomedical examples. When curves such as these are found, it is possible to use
either the logarithmic or perhaps a square-root transformation of X to achieve an
approximate straight line. The logarithmic transformation will tend to have a greater
effect than the square root. Both transformations have the effect of reducing large
values more than small values, and hence tend to make either curve a straight line. In
Figure 12.7, the arrows are drawn to show this reduction.
With statistical programs it is a simple matter to try various transformations and
then obtain the scatter diagram of the transformed X and original Y. If a logarithmic
transformation of X was successful, the resulting regression equation would be

Use of Transformations in Linear Regression

Y = a + b(log,, X)

It is preferable to try a transformation on X rather than Y since the resulting equation
is easier to interpret.
Additional information on the use of transformations in regression analysis may be
found in Lewis-Beck [ 19801, Afifi et al. [2004], Mickey et al. [2004], and Atkinson
[1985].